# The Fundamental Theorem of Calculus, Particle Motion, and Average Value

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1 The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt The Net Distnce the prticle trvels on the intervl from t = to t = b. If v( > on the intervl (, b), then it lso represents the Totl Distnce. b (3) v ( dt The Totl Distnce the prticle trvels on the intervl (, b), whether or not v( >. To be sfe, lwys do this integrl when sked to find totl distnce when given velocity. 1. The velocity of prticle tht is moving long the x xis is given by the function v( = 3t If the position of the prticle t t = 4 is 72, wht is the position when t = 2? b. Wht is the totl distnce the prticle trvels on the intervl t = to t = 7? Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 524

2 t 2. The velocity of prticle tht is moving long the x xis is given by the function 3. If the position of the prticle t t = 1.5 is 2.551, wht is the position when t = 3.5? v(.5e t 2. b. Wht is the totl distnce tht the object trvels on the intervl t = 1 to t = 5? The grph of the velocity, mesured in feet per second, of prticle moving long the x xis is pictured below. The position, p(, of the prticle t t = 8 is 12. Use the grph of v( to nswer the questions tht follow.. Wht is the position of the prticle t t = 3? b. Wht is the ccelertion when t = 5? c. Wht is the net distnce the prticle trvels from t = to t = 1? d. Wht is the totl distnce the prticle trvels from t = to t = 1? Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 525

3 The tble bove shows vlues of the velocity, V( in meters per second, of prticle moving long the x xis t selected vlues of time, t seconds.. Wht does the vlue of 18 V ( dt represent? b. Using left Riemnn sum of 6 subintervls of equl length, estimte the vlue of 18 V ( dt. Indicte units of mesure. c. Using right Riemnn sum of 6 subintervls of equl length, estimte the vlue of 18 V ( dt. Indicte units of mesure. d. Using midpoint Riemnn sum of 3 subintervls of equl length, estimte the vlue of 18 V ( dt. Indicte units of mesure. e. Using trpezoidl sum of 6 subintervls of equl length, estimte the vlue of 18 V ( dt. Indicte units of mesure. f. Find the verge ccelertion of the prticle from t = 3 to t = 9. For wht vlue of t, in the tble, is this verge ccelertion pproximtely equl to v (? Explin your resoning. Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 526

4 Interprettions nd Applictions of the Derivtive nd the Definite Integrl d [ AMOUNT] dx The rte t which tht mount is chnging For exmple, if wter is being drined from swimming pool nd R( represents the mount of wter, mesured in cubic feet, tht is in swimming pool t ny given time, mesured in hours, then R' ( would represent the rte t which the mount of wter is chnging. d [ R( ] R'( dx Wht would the units of R' ( be? b RATE AMOUNTOFCHANGE b In the context of the exmple sitution bove, explin wht this vlue represents: R' ( dt R( b) R( ). The tble given below represents the velocity of prticle t given vlues of t, where t is mesure in minutes. t minutes v( ft/minute 3. Approximte the vlue of v( dt using midpoint Riemnn Sum. Using correct units of mesure, explin wht this vlue represents. 25 b. Wht is the vlue of ( dt, nd using correct units, explin wht this vlue represents. 5 Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 527

5 The temperture of wter in tub t time t is modeled by strictly incresing, twice differentible function, W, where W( is mesured in degrees Fhrenheit nd t is mesured in minutes. Using the dt in the tble, estimte the vlue of W '(12). Using correct units, interpret the mening of this vlue in the context of this problem. 2 Use the dt in the tble to evlute W '( dt. Using correct units, interpret the mening of this integrl in the context of this problem. For 2 < t < 25, the function W tht models the wter temperture hs first derivtive given by the function W '(.4 t cos(.6. Bsed on this model, wht is the temperture of the wter t time t = 25? Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 528

6 A pn of biscuits is removed from n oven t which point in time, t = minutes, the temperture of the biscuits is 1 C. The rte t which the temperture of the biscuits is chnging is modeled by the.173t function B'( 13.84e. Find the vlue of B '(3). Using correct units, explin the mening of this vlue in the context of the problem. Sketch the grph of B' ( on the xes below. Explin in the context of the problem why the grph mkes sense. At time t = 1, wht is the temperture of the biscuits? Show your work. Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 529

7 A cylindricl cn of rdius 1 millimeters is used to mesure rinfll in Stormville. During the first 5 dys of 6-dy period, 3 millimeters of rinfll hd been collected. The height of wter in the cn is modeled by the function, S, where S( is mesured in millimeters nd t is mesured in dys for 5 < t < 6. The rte t which the height of the wter is rising is given by the function S '( 2sin(.3t ) Find the vlue of S '( dt. Using correct units, explin the mening of this vlue in the context of this problem. 1 At the end of the 6-dy period, wht is the volume of wter tht hd ccumulted in the cn? Show your work. The rte t which people enter n uditorium for concert is modeled by the function R given by 2 3 R( 138t 675t for < t < 2 hours; R( is mesured in people per hour. V.I.P. tickets were sold to 1 people who re lredy in the uditorium when the when the doors open t t = for generl dmission ticket holders to enter. The doors close nd the concert begins t t = 2. If ll of the V.I.P. ticket holders styed for the strt of the concert, how mny people re in the uditorium when the concert begins? Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 53

8 How hve we found Averge Velocity? Averge Vlue of Function How hve we found Averge Accelertion? If p(, v(, nd ( represent position, velocity nd ccelertion defined for ny time t, write n equivlent expression for ech of the following integrls bsed on the fundmentl theorem of clculus. 1 b 1 b b b ( dt v( dt To wht is this equivlent? To wht is this equivlent? The verge vlue of function, f(x), on n intervl [, b] is defined to be: Find the verge vlue of the function f ( x ) 3 x sin 2 x on the intervl 1 < x < 3. [Clcultor] Find the verge vlue of the function f(x) = 2 4x on the intervl 2 < x < 6. [Nonclcultor] Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 531

9 A ski resort uses snow mchine to control the snow level on ski slope. Over 24-hour period the 2 volume of snow dded to the slope per hour is modeled by the eqution t S( 24 t sin. The rte t 14 which the snow melts is modeled by the eqution t M ( 1 8cos. Both S( nd M( hve units of 3 cubic yrds per hour nd t is mesured in hours for < t < 24. At time t =, the slope holds 5 cubic yrds of snow.. Compute the totl volume of snow dded to the mountin over the first 6-hour period b. Find the vlue of M ( dt nd M ( dt 6 represents in the context of this problem.. Using correct units of mesure, explin wht ech c. Is the volume of snow incresing or decresing t time t = 4? Justify your nswer. d. How much snow is on the slope fter 5 hours? Show your work. e. Suppose the snow mchine is turned off t time t = 1. Write, but do not solve, n eqution tht could be solved to find the time t = K when the snow would ll be melted. Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 532

10 23 AB #6 Prt b Dily Lessons nd Assessments for AP* Clculus AB, A Complete Course Pge 533

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